You can get some of what you want, but most will be safely locked behind a login and therefore as difficult to access as it would be illegal! Depending on how busy the competitor site is this might be enough to get a good flavour:
Facebook : For fan numbers just check them out on facebook, which will give you an instant impression of how active they are there. For share/like metrics you can use opensiteexplorer.org (Pro members only though I think), although that is on a per URL basis rather than sitewide.
For Twitter you can again get their follower numbers just by visiting their profile and the share data for any URL though OSE or other tools (I quite like Raven for this).
For both of the above the OSE data based on the home page and a couple of sample URLs might be enough. If you don't mind a bit of coding you can also get this data from the APIs directly.
For analytics data you can find the UA number easy enough - it is there in plain text in the source code of every page. However by itself it is not useful at all and certainly won't give you access to someone elses data.
You best bet for that is to use one of the sampling services - or better still more than one! These use a sample of user data to give an indication of likely metrics. Do though bear in mind that they are all wrong! Compare them with your own site data and you will see that they are all wildly inaccurate. However they are still better than a blind guess and the busier the target site the more accurate they tend to be. Some useful places to look for this are (in no particular order)
There are others, but that should get you started.